X₁₄₉₃

NAPOLEON CROSSSUM

X₁₄₉₃ = midpoint of X₅₄ and X₁₉₅

5, 6, 7 3, 4, 5 Kepler Kimberling

X₁₄₉₃ (NAPOLEON CROSSSUM) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {54, 195}

• Related Centers: X₅₄ (KOSNITA POINT), X₁₉₅ (-CEVA CONJUGATE OF X₃)

Trilinears: \(f(A,B,C) : f(B,C,A) : f(C,A,B)\), where \(f(A,B,C) = \csc(B + \pi /6)\csc(C - \pi /6) + \csc(C + \pi /6)\csc(B - \pi /6)\)

Barycentrics: \((\sin A)f(A,B,C) : (\sin B)f(B,C,A) : (\sin C)f(C,A,B)\)

Search Value (6, 9, 13): \(-12.672820025801\)

▸ Alternate Coordinates1 form

• Trilinears: \(g(A,B,C) : g(B,C,A) : g(C,A,B)\), where \(g(A,B,C) = (3 \sin^{2} A - \cos^{2} A)(3 \sin B \sin C - \cos B \cos C)\)

▸ Incident Central Lines7 lines

{3, 54}, {5, 539}, {49, 143}, {110, 1173}, {113, 137}, {141, 575}, {206, 576}

▸ Functional Relations7 relations

• X₁₄₉₃ = midpoint of X₅₄ and X₁₉₅

• X₁₄₉₃ = isogonal conjugate of X₁₄₈₇

• X₁₄₉₃ = complement of X₃₅₁₉

• X₁₄₉₃ = X₁₁₀-Ceva conjugate of X₁₅₁₀

• X₁₄₉₃ = crosspoint of X₆₁ and X₆₂

• X₁₄₉₃ = crosssum of X₁₇ and X₁₈

• X₁₄₉₃ = orthocenter of pedal triangle of X₅₄

Notes

A construction of X₁₄₉₃ is given by Antreas Hatipolakis and Angel Montesdeoca at 24179.